THE  TREND  OF  THE  CONTENT  OF  FIRST-YEAR  ALGEBRA  TEXTS 


BY 

ELINOR  BERTHA  FLAGG 

B.  S.  University  of  Illinois,  1921 


THESIS 

SUBMITTED  IN  PARTIAL  FULFILLMENT  OF  THE  REQUIREMENTS 
FOR  THE  DEGREE  OF  MASTER  OF  SCIENCE  IN  EDUCATION 
IN  THE  GRADUATE  SCHOOL  OF  THE 
UIVERSITY  OF  ILLINOIS 
1922 


URBANA,  ILLINOIS 


£ 53  

UNIVERSITY  OF  ILLINOIS 

THE  GRADUATE  SCHOOL 


JUriG  cL  , _JQ2<- 


1 HEREBY  RECOMMEND  THAT  THE  THESIS  PREPARED  UNDER  MY 
SUPERVISION  by  Elinor  Bertha  Flagg 

ENTITLED The. .Trend of  the  Content  of  First.- Year  Algebra- 

Texts — 

BE  ACCEPTED  AS  FULFILLING  THIS  PART  OF  THE  REQUIREMENTS  FOR 
THE  DEGREE  OF  Master  Qf__SclerLQa_J.n  Educ atl.on 


Recommendation  concurred  iiP 


Committee 


on 


Final  Examination* 


•Required  for  doctor’s  degree  but  not  for  master’s 


A i 

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Digitized  by  the  Internet  Archive 
in  2015 


https://archive.org/details/trendofcontentofOOflag 


TABLE  OF  CONTENTS 

Chapter  Page 

I Introduction  1 

II  The  Problem  and  the  Method  of 

this  Investigation  15 

III  The  Content  of  Twelve  First  Year 

Algebra  Texts  E4 

IV  Summary • The  Trend  of  the  Content  56 


Chapter  I 
Introduction 

Discussions  of  the  secondary  school  curriculum  of  today 
never  fail  to  provoke  lengthy  debates  concerning  mathematics. 
These  debates  center  around  the  following  questions.  What  topics 
and  processes  should  be  taught?  to  whom?  how  much?  and  why? 
Such  questions  have  become  increasingly  important  because  of  the 
interests  and  capacities  of  the  pupils  now  entering  high  school. 
Formerly  students  attending  secondary  schools  were  a highly  se- 
lected group.  Their  purpose  in  attending  these  schools  was  to 
prepare  for  college  and  courses  were  planned  to  meet  the  demands 
which  the  college  made  for  entrance.  The  function  of  the  modern 
high  school,  however,  cannot  be  stated  as  simply  as  this.  Older 
educators  coped  with  a simple  industrial  situation  while  the 
modern  educator  faces  an  extremely  complex  situation.  The  watch- 
word of  the  modern  high  school,  as  Johnston  quotes,  is  M equal 
opportunity  for  all  the  children  of  all  people” • * 

This  means  that  today  all  pupils  of  the  high  school  age 
have  a right  to  demand  an  education;  and  in  this  day  of  de- 
mocracy **  America  demands  that  we  educate  the  whole  group. 

* Johnston, G.  H.  — Modern  High  School,  Chapter  I,  P.  10 

Jessup,  W.  A.  --  "The  Greatest  Need  of  the  Schools  --  Better 

Teaching.”  Journal  of  N.  E.  A.,  Vol.  X, 

No.  4,  Pp.  71  - 73,  April  1921 


2 

With  this  ideal  our  high  school  enrollment  necessarily  Includes 
pupils  from  many  walks  of  life . We  no  longer  have  only  hoys 
and  girls  of  exceptional  ability. 

Thorndike  points  out  * that  our  pupils  of  today  differ 
from  those  of  twenty-five  years  ago  in  their  experiences  and 
interests  and  in  their  capacities  and  abilities.  They  are  born 
into  a more  complex  social  and  industrial  organization.  In- 
dustries and  industrial  centers  have  enlarged;  more  children 
are  found  in  large  cities  where  they  are  experiencing  various 
modern  inventions.  The  automobile,  the  wireless,  and  innumer- 
able changes  in  factory  machinery  are  new  in  the  last  quarter 
of  a century. 

As  a result  of  these  changing  conditions  the  enrollment 
of  the  high  school  has  changed  both  quantitatively  and  qualita- 
tively. Thorndike  estimates  * that  the  number  of  high  school 
pupils  in  1918  was  six  times  that  in  1890,  while  the  population 
had  not  doubled  in  that  time.  He  also  shows  that  for  every  one 
hundred  children  who  reached  the  age  of  fourteen  there  were 
about  three  and  one-half  times  as  many  entering  high  school  in 
1918  as  in  1 890 ; therefore  we  may  say  that  where  one  pupil  in 
ten  entered  high  school  in  1890,  in  1918  one  pupil  in  three 
entered.  The  interests  of  these  pupils  were  more  varied  in 
1918  than  in  I89O;  some  came  because  the  school  lav/  compelled 
them  to  stay  in  school  up  to  the  age  of  fourteen  or  sixteen:  ** 

* Thorndike,  E.  L.  — "Changes  in  the  Quality  of  the  Pupils 

Entering  High  School,"  School  Review, 

Vol . XXX,  No.  5,  Pp  355-359,  May  1922 
**  U.  S.  Bulletin,  Bureau  of  Education.  "Part-Time  Education  of 

Various  Types,  No.  5,  1921 


I 

. 


. • 


. 

. 


formerly  these  pupils  would  have  gone  to  work  instead  of  entering 
high  school.  Others  come  to  prepare  for  college,  or  simply  to 
"go  thru"  high  school. 

With  both  industrial  and  college  interests  in  the  high 
school,  the  value  of  the  traditional  college  preparation  courses 
for  all  pupils  came  to  be  questioned.  It  was  felt  that  the  new 
interests  present  in  the  high  school  should  be  recognized,  that 
something  should  be  changed.  Hester,  * in  an  article  on  "Eco- 
nomics in  the  Course  in  Mathematics  from  the  standpoint  of  the 
High  School,"  says,  "We  all  know  there  are  topics  in  the  high 
school  which  experience  does  not  justify  spending  time  on,  but 
what  they  are  we  are  not  quite  agreed."  As  a result  a discontent 
arose  concerning  the  traditional  mathematics  course.  He  goes  on 
to  say  that  the  outstanding  causes  of  dissatisfaction  with  our 
mathematics  courses  are  two:  (1)  the  general  attack  upon  the 

doctrine  of  formal  discipline  and  (2)  the  rapid  increase  of  in- 
dustrial and  practical  education  in  the  elementary  and  secondary 
schools.  These  attacks  indicated  that  mathematics,  of  long 
standing  as  the  best  example  of  a logically  organized  subject, 
should  be  psychologized,  reconstructed,  and  reorganized  to  place 
greater  emphasis  upon  the  life-mathematics  needed  by  the  average 
pupil.  The  interests  and  capacities  of  the  pupil  should  be  given 
attention. 

Educators  are  offering  various  suggestions  for  improve- 
ment. There  are  those  who  would  not  have  courses  in  mathematics 

* Hester,  Frank  0.  --  "Economics  in  the  Course  in  Mathematics 

from  the  Standpoint  of  the  High  School," 
School  Science  and  Mathematics,  Vol.  XIII 
PP.  751  - 757 


V. 

* 


. 

' 


4 


changed.  This  conservative  group  would  keep  the  traditional 
content  and  would  find  the  hope  of  improvement  in  a more  adequate 
preparation  of  teachers.  One  radical  group  would  teach  only  those 
topics  and  processes  that  the  pupil  will  actually  use.  Another 
radical  group  would  eliminate  mathematics  from  the  required 
courses. 

Still  another  attempt  at  change  is  noted  in  the  move- 
ment for  "unified  mathematics" . Two  outstanding  publications 
along  this  line  are  Breslich's  three  volumes,  First-Year,  Second- 
Year  and  Third- Year  Mathematics,  and  Rugg  and  Clark's  "Funda- 
mentals of  High  School  Mathematics" . 

Breslich's  Course.  These  three  hooks  are  based  on  the 
viewpoint  that  there  are  fundamental  operations  in  geometry  and 
algebra  --  distinct  bodies  of  material  in  the  two  traditional 
subjects  which  ought  in  some  way  to  be  merged  into  a coherent 
course.  "When  the  various  branches  of  mathematics  are  treated 
as  separate  subjects,"  Breslich  says ,*" there  is  a tendency  for 
each  to  take  on  the  rigid  form  of  the  final  science.  This  tends 
inevitably  to  a certain  formalism  in  mode  of  presentation.  Such 
formalism  is  not  the  best  method  for  the  beginner.  Correlation 
helps  to  avoid  excessive  formalism."  Concerning  the  presenta- 
tion of  material  Breslich  adds,  "For  correlated  mathematics  it 
is  relatively  easy  to  adopt  a method  of  approach  that  is  induc- 
tive. In  Geometry  the  peculiar  properties  of  the  appropriate 
figures  as  studied  and  the  results  are  then  combined  into  a 

* Breslich,  E.  R.  — First  Year  Mathematics  (1916)  Author's 

preface  - ~ 


: 


. 


5 


theorem.  This  brings  about  an  easier  and  a much  better  under- 
standing  than  a beginner  can  obtain  from  a logical  proof.  Axioms 
usually  assumed  to  be  self-evident  are  in  the  following  pages 
illustrated  in  order  to  make  their  mea.ning  apparent  and  vital. 

Not  until  toward  the  end  of  the  geometry  of  the  first  year  does 
the  demonstration  take  the  form  of  a logical  proof.  Even  then 
the  method  of  proof  is  informal,  the  aim  being  to  convince  the 
student  by  the  truth  of  the  theorem,  to  enable  him  to  U3e  a 
theorem  to  establish  other  facts,  and  to  prepare  him  gradually 
for  the  formal  geometry  of  the  second  course.  Algebra  is  intro- 
duced as  a natural  means  of  expressing  facts  about  numbers  and 
gradually  becomes  a symbolic  language  especially  well  adapted  to 
stating  the  conditions  of  a problem  in  a natural  and  helpful 
way.  The  growing  difficulty  and  complexity  of  problems  then  lead 
to  the  necessity  of  learning  how  to  manipulate  algebraic  symbols. 
The  symbolism  of  algebra  thus  becomes  a highly  clarifying  instru- 
ment of  problem  analysis  and  problem  solving.  The  laws  of  alge- 
bra are  carefully  illustrated  thus  avoiding  the  danger  of  symbol- 
juggling  without  insight  into  the  real  meaning. 

"There  are  certain  processes  which  belong  together 
logically  but  which  should  be  separated  in  treatment  because 
they  make  difficulties  for  the  beginner.  Hence,  wherever  the 
processes  are  not  needed  as  instruments  of  instruction,  they  are 
taught  separately;  for  example,  the  meaning  of  positive  and 
negative  numbers,  the  laws  of  signs,  and  the  operations  with 
positive  and  negative  numbers  are  not  studied  until  the  pupil 
has  become  thoroughly  familiar  with  unsigned  literal  numbers  and 


. 

. 


r- 

, 


. 


- 


- 


, 


6 


with  the  operations  of  laws  of  such  literal  numbers. " 

The  material  to  be  given  in  Rugg  and  Clark’s  text  was 
determined  by  a thorough  investigation  on  "Scientific  Method  in 
the  Reconstruction  of  Ninth  Grade  Mathematics" . 

Rugg  and  Clark’s  Investigation  of  Ninth  Year  Mathe- 
matics. This  study  shows  that  one- third  of  our  instructional 
attention  in  the  first -year  algebra  is  devoted  to  formal  types 
of  material.  One-sixth  of  the  formal  examples  of  text  books  are 
based  upon  the  four  fundamental  operations,  and  another  sixth  of 
the  text  is  devoted  to  special  products  and  factoring.  One- 
fourth  of  the  entire  problem  material  is  devoted  to  the  equation 
"which  nearly  all  mathematics  teachers  embrace  as  the  central 
operation  of  algebra" . Their  analysis  of  the  use  of  current 
ninth-grade  mathematics  in  other  high  school  subjects  and  in  oc- 
cupational activities  shows  that  it  is  impossible  to  defend  the 
large  amount  of  attention  to  formal  material  based  on  the  funda- 
mental operations  and  absolutely  impossible  to  defend  this  em- 
phasis upon  special  products  and  factoring,  on  the  basis  of  use. 
It  is  their  opinion  that  more  than  one  half  of  the  subject  matter 
that  is  taught  will  never  be  used  by  the  vast  majority,  neither 
in  the  high  school,  in  occupational  or  other  life  activities 
beyond  the  school,  or  even  by  that  fraction  of  percent  of  the 
population  that  engages  in  the  various  scientific  professions. 
Desiring  to  have  mathematics  taught  for  greater  usefulness  they 
recommend  the  following  mathematical  creed:  "to  develop  in  the 

pupil  the  ability  to  use  intelligently  the  most  powerful  devices 


. 

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7 

of  quantitative  thinking:  the  equation,  the  formula,  the  graph 
and  the  properties  of  the  more  important  space  forms." 

Fundamental  to  the  improvement  of  the  present  situation 
Rugg  and  Clark  enumerate  these  two  important  steps: 

(1)  The  thorough  overhauling  of  the  course  of  study  in 
mathematics,  the  elimination  of  non-essential  material,  the  addi- 
tion of  the  types  of  real  mathematics  not  now  a part  of  the  ninth 
grade  course,  and  the  construction  of  a continuous  mathematical 
course,  worked  out  around  two  basic  principles,  one  mathematical 
and  the  other  psychological. 

(2)  The  improvement  of  methods  of  teaching  mathematics 
to  ninth  grade  students.  Ideally  this  demands  better  training  of 
mathematics  teachers.  For  this  improvement  we  need  a new  type 

of  textbook,  a wordy  textbook,  a real  applied  psychology  for  the 
teacher,  standardized  tests  with  which  to  check  up  at  intervals 
throughout  the  year  the  results  obtained  from  instruction,  and 
practice  devices  perfecting  the  former  skills. 

Rugg  and  Clark’s  text,  bearing  the  title  "Fundamentals 
of  High  School  Mathematics",  includes  first,  selected  material 
which  is  now  in  the  traditional  first  year  course,  and  second, 
much  material  which  is  either  not  taught  at  all  to  high  school 
students  or  is  taught  to  only  a limited  portion  in  some  advanced 
year.  The  tv/o  principles  controlling  their  selection  of  subject 
matter  we re  social  worth  and  thinking  value.  The  new  course  of 
study  based  on  the  criterion  of  social  worth  must  include,  Rugg 
and  Clark  say,  "training  in  (a)  the  use  of  letters  to  represent 


• 

< 

. 

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8 


numbers;  (b)  the  use  of  the  simple  equation;  (c)  the  construction 
and  evaluation  of  formulas;  (d)  the  finding  of  unknown  distances 
by  means  of  (l)  scale  drawings,  (2)  the  principle  of  similarity 
in  triangles,  (3)  the  use  of  the  properties  of  the  right  triangle 
(ratios  of  the  sides  as  in  the  Hypotenuse  Rule  and  in  the  cosine 
and  the  tangent  of  an  angle) ; (e)  the  preparation  and  use  of 
statistical  tables  and  graphs  to  represent  and  compare  quantities 
(this  includes  the  group  of  elementary  statistical  measures.)” 

The  writers  of  the  text,  however,  are  of  the  group  that 
regard  "thinking"  outcomes  as  coordinate  in  importance  with  the 
more  common  social  utility.  The  entire  course  has  been  organized 
around  the  central  core  of  "problem-solving".  "Even  the  purely 
formal  materials  themselves,"  they  say,  "have  been  so  organized, 
wherever  possible,  as  to  provide  an  opportunity  for  real  thinking 
and  not  mere  habit  formation."  Also,  an  attempt  has  been  made  to 
build  the  course  in  such  a way  as  to  "contribute  constantly  to 
ability  in  the  expression  and  determination  of  relationship." 

In  this  book  the  content  of  the  course  was  selected  to 
i satisfy  "rigorous  criteria  of  either  social  worth  or  definitely 
established  thinking  value,  or  both".  * The  presentation  of  ma- 
terial of  this  type  led  to  "(*)  vast  economy  of  time  by  excluding 
non-essential  operations  and  forms;  (2)  introduction  of  new  ma- 
ter ial  not  commonly  attempted  or  successfully  taught  in  the  first 
year  course,  for  example,  (a)  the  use  of  statistical  measures, 
tables,  and  graphs  to  represent  and  compare  quantities,  (b)  the 


* Rugg  and  Clark,  Fundamentals  of  High  School  Mathematics  (19*9) 

Preface  - - 


. 


. 


.• 


“ 


9 


organization  of  a whole  course  about  the  central  theme  of  relation- 
ship and  the  systematic  organisation  of  three  methods  represent- 
ing  and  determining  relationship  --  the  graphic  method,  the  tabu- 
lar method,  and  the  equational  or  formula  method,  (c)  systematic 
teaching  methods  of  indirect  measurement,  for  example,  the  find- 
ing the  unknown  distance  by  scale  drawings,  similar  triangles, 
and  the  use  of  properties  of  the  right  triangle." 

The  Investigation  of  the  National  Committee.  Another 
attempt  to  reorganize  high  school  mathematics  is  indicated  by  the 
National  Committee  on  Mathematical  Requirements  in  a report  on 
"The  Reorganization  of  the  First  Courses  in  Secondary  School 
Mathematics."  This  committee,  working  under  the  auspices  of  the 
Mathematical  Association  of  America,  published  this  report  which 
was  preliminary  in  February  1920.  A full  report  bearing  the 
title  "The  Reorganization  of  Mathematics  in  Secondary  Education" 
was  published  the  following  year.  The  material  contained  in  the 
preliminary  report  is  of  primary  interest  to  this  thesis.  Since 
it  contains  all  the  material  basic  to  the  later  report  and  also 
the  material  on  which  the  problem  of  this  investigation  is 
founded,  a summary  of  only  the  preliminary  report  will  be  given 
here . 

The  National  Committee  suggest  the  most  desirable  train- 
ing to  be  included  in  ninth  and  tenth  year  mathematics,  taking 
secondary  school  mathematics  as  a whole,  for  they  say  "at  the 
present  stage  in  the  discussion,  agreement  as  to  what  should 
constitute  the  work  of  the  ninth  and  tenth  years  separately  may 


1 


I 

10 

. 

be  difficult  to  secure".  The  material  considered  desirable  was 
determined  by  two  fundamental  principles. 

"(1)  The  primary  purposes  of  the  teaching  of  mathematics 
should  be  to  develop  those  powers  of  understanding  and  analyzing 
relations  of  quantity  and  of  space  which  are  necessary  to  a better 
appreciation  of  the  progress  of  civilization  and  a better  under- 
standing  of  life  and  of  the  universe  about  us,  and  to  develop 
those  habits  of  thinking  which  will  make  these  powers  effective 
in  the  life  of  the  individual. 

"(2)  The  courses  in  each  year  should  be  so  planned  as 
to  give  the  pupil  the  most  valuable  mathematical  information  and 
training  which  he  is  capable  of  receiving  in  that  year  with  little 
reference  to  the  courses  which  he  may  or  may  not  take  in  succeed- 
ing years . 

"The  general  topics  under  which  all  details  should  fall 
are  the  formula,  graphic  representation,  the  equation,  measure- 
ment and  computation,  congruence  and  similarity,  demonstration. 

To  impart  these  ideas,  ’practical1  problems,  real  to  the  pupil, 

I 

must  be  connected  with  his  experience  and  interest.  To  Unify  the 

i 

course,  the  idea  of  functional  relations  is  sufficient. 

"Continued  emphasis  throughout  the  course  must  be 
placed  on  the  development  of  power  in  applying  ideas,  processes, 
and  principles  to  concrete  problems  rather  than  to  acquisition 
Of  mere  facility  or  skill  in  manipulation.  On  the  side  of  alge- 
bra, the  ability  to  analyze  a problem,  to  formulate  it  mathematical- 
ly, and  to  interpret  the  result  must  be  dominant  aims.  Drill  in 
algebraic  manipulation  should  be  limited  to  those  processes  and 


* 

. 


t 

. 


. 


, 

11 


to  the  degree  of  complexity  required  for  a thorough  understanding 
of  principles  and  for  probable  applications  either  in  common  life 
or  in  subsequent  mathematics.  It  must  be  conceived  throughout 
as  a means  to  an  end,  not  as  an  end  in  itself.  Within  these 
limits  skill  in  algebraic  manipulation  is  important  and  drill  in 
this  subject  should  be  extended  far  enough  to  enable  students  to 
carry  out  the  fundamentally  essential  processes  accurately  and 
expeditiously.'* 

Minimum  essentials  recommended  by  the  National  Committee. 
This  list  of  topics  and  processes  is  considered  by  the  committee 
to  be  the  essential  part  of  high  school  algebra.  These  topics 
and  processes  are  those  contributing  to  the  development  of  the 
“basic  principles"  quoted  above.  * 

(1)  The  formula,  its  meaning  and  use • 

a.  As  a concise  language.  - 

b.  As  a shorthand  rule  for  computation. 

c . As  a general  solution. 

d.  As  an  expression  of  the  dependence  of 

one  variable  on  another  variable. 

(2)  The  graph  and  graphic  representations  in  general  - 

their  construction  and  interpretation: 

a.  As  a method  of  representing  facts  (sta- 

tistical, etc.) 

b.  As  a method  of  representing  dependence. 

c.  As  a method  of  solving  problems. 

(3)  Positive  and  negative  numbers  --  Their  meaning 

and  use : 

a.  As  expressing  both  magnitude  and  one  of 

two  opposite  directions  or  senses. 

b.  Their  graphic  representation. 

c.  The  fundamental  operations  applied  to  them 

* Much  of  the  same  type  of  information  is  given  concerning  geometry 
as  has  here  been  summed  up  regarding  algebra,  but  since  this  in- 
vestigation is  concerned  primarily  with  algebraic  material,  it  was 
not  considered  essential  that  the  geometric  part  be  reported. 


12 


(4)  The  equation  — its  use  in  solving  problems: 

a.  The  linear  and  the  quadratic  equation 

in  one  unknown,  their  solution  and 
applications . 

b.  Equations  in  two  variables. 

1 . As  expressing  a functional  relation, 

with  numerous  concrete  illustrations 

2.  As  making  possible  the  determination 

of  the  unknowns. 

c.  Ratio,  proportion,  variation. 

(5)  Algebraic  technique: 

. . a.  The  fundamental  operations. 

b.  Factoring.  The  only  cases  that  need  con- 

sideration are  monomial  factors,  the 
difference  of  two  squares,  the  square 
of  a binomial,  and  trinomials  of  the 
second  degree  that  can  easily  be  factored 
by  trial. 

c.  Fractions. 

d.  Exponents  and  radicals: 

1.  Laws  for  positive  integral  exponents. 

2.  Radicals  should  be  confined  to  the 

simplification  of  expressions  the 
form  a2b  and  a and  to  the  numeri- 
c b 

cal  evaluations  of  simple  ex- 
pressions involving  the  radical 
sign. 

3»  Extracting  the  square  roots  of 
numbers . 

Optional  topics.  When  schools  can  cover  satisfactorily 
the  work  suggested,  the  Committee  favors  introducing  the  follow- 
ing topics  and  processes  earlier  than  is  now  customary: 

(1)  Meaning  and  use  of  fractional  and  negative 

exponents . 

(2)  Use  of  logarithms  and  other  simple  tables. 

(3)  Use  of  the  slide  rule. 

(4)  Simple  work  in  arithmetic  and  geometric  progressions 

(on  account  of  their  importance  in  financial  and 
in  scientific  thinking.) 

(5)  Simple  problems  involving  combinations,  and  prob- 

ability (on  account  of  their  frequent  occurrence 
in  daily  life  and  their  thought  provoking  qualities) 

(6)  Elementary  ideas  concerning  statistics. 


. 


13 

The  Committee  recommends  that  all  topics  and  processes 
which  do  not  directly  contribute  to  the  development  of  the  powers 
mentioned  in  the  fundamental  principle  should  be  eliminated  from 
the  course. 

Accordingly  the  list  of  topics  and  processes  quoted  be- 
low is  recommended  "to  be  omitted",  from  the  work  of  the  first 
two  years i 

(1)  Highest  common  factor  and  lowest  common  multiple, 

except  in  the  simplest  cases  involved  in  the 
addition  of  simple  fractions. 

(2)  The  theorems  on  proportion  relating  to  alteration, 

inversion, composition  and  division. 

(3)  Literal  equations,  except  such  as  appear  in  common 

formulas,  such  as  may  be  necessary  in  the  deriva- 
tion of  formulas,  the  discussion  of  geometric 
facts,  or  to  show  how  needless  computation  may  be 
avoided. 

(4)  Radicals  except  as  indicated  elsewhere. 

(5)  Extraction  of  square  roots  of  polynomials. 

(6)  Cube  root. 

(7)  Theory  of  exponents. 

(8)  Simultaneous  quadratic  equations  in  more  than  two 

unknowns . 

(9)  Pairs  of  simultaneous  quadratic  equations. 

(10)  The  theory  of  quadratic  equations  (remainder  and 

factor  theorems,  etc.) 

(11)  Binomial  theorem. 

(12)  Arithmetic  and  geometric  progressions. 

(13)  Theory  of  imaginary  and  complex  numbers. 

( 14)  Radical  equations,  except  such  as  arise  in  dealing 

with  elementary  formulas. 

(15)  All  equations  of  degree  higher  than  the  second. 


14 


History  and  biography,  the  committee  says,  should  he 
used  incidentally  for  emphasizing  the  idea  that  mathematics  has 
been,  and  is  a growing  science. 

Summary . These  investigations  and  texts  show  a vast 
change  from  the  time  when  high  school  algebra  was  taught  only  to 
fulfill  college  entrance  requirements.  They  sho?/  an  attempt  to 
meet  present  day  conditions;  along  with  preparing  pupils  for 
college  the  high  school  today  trains  many  boys  and  girls  who  enter 
industry  after  one  year  in  the  high  school.  These  pupils  need 
especially  the  topics  and  processes  that  will  help  them  in  their 
work.  Recognizing  this  need  for  connecting  subject  matter  with 
life,  that  is  a vital  school  problem  today,  these  investigations 
recommend  modifications  of  the  traditional  formal  mathematics, 
and  the  text  books  of  Breslich,  and  Rugg  and  Clark  give  evidence 
of  changes  from  the  formal  teaching  of  algebra  and  geometry. 


* 

' 


. 


X > 


. 

. ■ 

■ . 

. 


Chapter  IX 

The  Problem  and  the  Method  of  this  Investigation 

Professor  L.  V.  Koos  * in  an  investigation  concerning 
secondary  school  mathematics  points  out  that  textbooks  dominate 
the  content  and  organization  of  courses  in  mathematics.  Results 
from  responses  to  questionnaires  concerning  the  degree  to  which 
high  school  teachers  followed  the  textbooks  showed  the  following 
deviations  from  the  plan  of  the  text: 

Deviations  from  the  plan  of  the  text.  Number  of  texts. 


Omissions 
Additions 
Shifts  of  order 
None 

No  answer 


21 

6 

21 

42 

19 


Of  the  ninety  responses  received,  sixty-three  reported  no  change 
in  the  textbook  material  or  merely  changed  the  order  of  intro- 
duction of  topics,  twenty-one  omitted  part  of  the  text,  and  only 
six  high  school  teachers  added  to  the  content  presented  in  the 
text . 

Professor  Koos*  investigation  also  shows  that  elementary 
algebra  is  almost  always  a first-year  high  school  subject.  ’’Plane 
geometry,”  he  says,  ” is  markedly  a second-year  subject,  but  is 
reported  in  some  schools  in  the  third  year,  or  in  the  latter  half 


Koos,  L.  V.  — 


’’The  Administration  of  Secondary-School  Units 
(Supplementary  Educational  Monographs,  Vol.  I, 
No.  3,  Whole  No.  3)  Chicago:  The  University 
of  Chicago  Press,  July  1917. 


16 

of  the  second  year  and  the  first  half  of  the  third.  Advanced  alge- 
bra appears  most  commonly  in  the  third  and  fourth  years,  but  in  a 
few  schools  in  the  second.  Solid  geometry  appears  in  the  third 
and  fourth  years  and  trigonometry  in  the  fourth  year.” 

Hence,  it  is  concluded  (1)  that  high  school  teachers 
cling  closely  to  the  text  for  the  material  taught  in  algebra 
courses,  and  (2)  that  algebra  is  essentially  a first-year  high 
school  subject. 

The  report  of  the  National  Committee  on  Mathematical 
Requirements  is  the  most  recent  study  dealing  with  present  day 
standards  in  secondary  school  mathematics.  Because  it  is  modern, 
the  topics  and  processes  recommended  therein  have,  for  the  purpose 
of  this  investigation,  been  considered  as  the  criterion  for  our 
modern  high  school;  hence  for  our  present  purpose,  a text  which 
meets  the  recommendations  of  the  National  Committee  is  meeting  the 
requirements  of  the  modern  high  school. 

It  is  the  problem  of  this  investigation  to  determine 
whether  the  trend  of  the  content  of  algebra  texts  is  toward  meet- 
ing the  requirements  of  the  modern  high  school  as  they  are  stated 
in  the  recommendations  of  the  National  Committee.  The  trend  of 
the  content  may  be  known  by  a historical  study  of  the  material 
contained  in  a number  of  text3,  and  a comparison  of  the  content 
of  texts  through  a number  of  years  with  the  content  recommended 
by  the  Committee,  will  indicate  whether  the  trend  is  toward  the 
recommendations  of  the  Committee  or  not. 

Statement  of  Procedure . For  the  purpose  of  obtaining 


information  relative  to  the  trend  of  the  content  of  algebra  texts. 


17 


twelve  one-volume  first-year  algebra  texts  representing  a period 
of  thirty-nine  years  were  selected  for  study.  These  were  not 
chosen  in  any  scientific  way;  they  were  selected  because  of  wide 
usage  and  with  the  advice  of  certain  persons  whose  acquaintance 
with  the  field  of  secondary  mathematics  has  extended  over  a number 


The 

texts  and  the 

date  of  publication  are  as  follows: 

1 . 

Wentworth  

Elements  of  Algebra 

1881 

2. 

Milne  

- High  School  Algebra 

1892 

2 • 

Wells  

- Essentials  of  Algebra 

1827 

4. 

Wentworth  

First  Steps  in  Algebra 

1898 

5. 

Wells  

- Algebra  for  Secondary 
Schools 

1906 

6 . 

Young  and  Jackson  - First  Course  in 

Elementary  Algebra 

1908 

7. 

Hawkes,  Luby 

and  Touton  - First  Course 
in  Elementary  Algebra 

1910 

8. 

Wells  and  Hart  --  First  Year  Algebra 

1912 

9. 

Milne  

--  First  Year  Algebra 

1915 

10. 

Slaught  and  Lennes  - Elementary 

Algebra 

1915 

1 1 . 

Hawke  s , Luby 

and  Touton  - First  Course 
in  Algebra 

1917 

12. 

Durell  and  Arnold  - First  Book  in 

Algebra 

1920 

The  wide  use  of  Wentworth,  Milne,  Wells,  and  Hawkes, 
Luby  and  Touton  is  evident  from  the  fact  that  they  have  been  re- 
vised and  with  the  exception  of  Wentworth's  text  they  are  still 
in  use.  Young  and  Jackson  also  is  still  in  use.  Slaught  and 
Lennes  is  a comparatively  new  text,  but  it  has  been  revised  once 


' 


18 


and  hag  found  wide  usage.  Durell  and  Arnold  was  selected  as  the 
newest  first-year  algebra. 

Method  of  Examining;  Texts.  In  Chapter  I the  topics 
"to  be  included"  and  those  "to  be  omitted"  were  quoted  from  the 
report  of  the  National  Committee,  and  it  is  not  necessary  that 
they  be  re-quoted  here.  It  is  sufficient  to  state  here  the  studies 
based  on  these  two  lists  of  topics.  A study  was  made  of 

(1)  Topics  and  space  included  which  the  Committee 

recommended  "to  be  included", 

(2)  Topics  and  space  included  which  the  Committee 

recommended  "to  be  omitted", 

(3)  The  percent  of  drill  exercises,  or  examples,  and 

also  the  percent  of  verbal  exercises  or  problems, 

(4)  The  space  given  to  history  and  biography. 

For  each  study  it  was  necessary  to  set  up  criteria  for 
each  topic  in  order  that  a uniform  comparison  might  be  drawn. 

Some  topics  may  be  definitely  placed  under  one  of  the  headings  in 
either  the  list  "to  be  included"  or  the  list  "to  be  omitted"  by 
merely  glancing  at  the  name  of  the  topic.  For  example,  no  criteria 
is  essential  for  the  topic  "square  root  of  polynomials";  this 
heading  in  itself  places  the  topic.  Other  topics  of  this  kina, 
requiring  no  statement  of  criteria,  are  parenthesis,  fundamental 
operations,  positive  and  negative  numbers,  exponents  and  radicals, 
highest  common  factor  and  lowest  common  multiple,  binomial  theorem, 
cube  root,  and  progressions. 

But  merely  naming  a topic  or  process  does  not  always 
place  it  as  a topic  "to  be  included"  or  one  "to  be  omitted",  since 
it  is  frequently  found  that  one  subdivision  of  a topic  is 


19 


desirable  and  another  is  undesirable.  For  example,  under  "equa- 
tions" some  topics  are  "to  be  included"  and  others  are  "to  be 
omitted" . It  is  necessary  to  divide  this  topic  into  those  sug- 
gested by  the  Committee:  linear  and  quadratic  equations  in  one 

unknown,  and  linear  and  quadratic  equations  in  two  unknowns  as 
approved  by  the  Committee,  are  "topics  to  be  included";  while 
literal  equations  (except  such  as  appear  in  common  formulas), 
simultaneous  equations  in  more  than  two  unknowns,  pairs  of  simul- 
taneous quadratic  and  radical  equations  are  "topics  to  be  omitted". 
As  illustrations  of  both  types  of  exercises,  the  following 
examples  are  given: 

Topics  "to  be  included": 

linear  in  one  unknown, 
quadratic  in  one  unknown, 
linear  in  two  unknowns, 

quadratic  in  two  unknowns, 

and 

literal  equations  that  are 
formulas, 

radical  equations  that  are 

formulas, 

Topics  "to  be  omitted": 

literal  equations,  as 

simultaneous  equations  in 
more  than  two  unknowns,  as 


ICx  - 5=  3x  +•  30 

x2  ~ 5x  = 24 

(3x  -t  y - 1 1 
(5x  - y - 13 

(xf  y ^ -3 
( xy  — 54  - 

a-  -h  b2  = - 1 
V — lwh 
5 - 1/2  gt2 

2x  f a - 7a  - x 

:!2x  - yfz  - 5 
i3x  +■  2y  - 3z  — 7 
Ax  - 3 y - 5z  — -3 


20 


pairs  of  simultaneous  quadratics 
of  either  of  these  two  types: 
(1)  two  equations  of  the 


second  degree,  as 


( 2 ) one  equation  of  the  second 
degree , a 3 


radical  equations,  as 


3 - T =“\/5c 


Other  topics  "to  he  included”  and  "to  he  omitted" 
which  are  not  entirely  clear  in  themselves  and  the  criteria  for 
these  topics  and  processes  are  as  f ollows : 

Formula,  "Formula"  is  considered  "to  he  included" 
where  the  topic  is  specifically  given  or  where  it  is  implied 
and  various  formulas  are  given  "as  a concise  language,  as  a 
shorthand  rule  for  computation,  as  a general  solution,  or  as  an 
expression  of  the  dependence  of  one  variable  on  other  variable s" . 
For  example,  material  of  this  type  should  he  included:  (1)  Find 

the  formula  for  the  area  of  a rectangle;  (2)  Express  this  rule 
in  words : A=  tt  r 

Graph . "Graph"  is  considered  to  include  all  material 
on  the  graph  and  graphing  which  is  given  anywhere  in  the  text. 
Illustrations  of  various  types  of  material  given  under  this  topic 
are  as  follows:  (1)  Determine  the  length  of  the  rivers  in  Fig- 

ure R (given  in  the  text^ ; (2)  Draw  graph  of  the  equation 
L ^ 1 ; (3)  G-et  the  temperature  readings  in  your  own  school 

district  tomorrow  and  draw  the  graph!  (4)  Draw  the  graph  of 

y . x Select  at  least  four  negative  and  four  positive  values 

2 5 ~ 1 * 


«.  ' 


. 


' . 


t 


...  v*  i. . . . 


21 


of  x,  and  from  them  determine  the  corresponding  values  of  y. 
What  sort  of  graph  do  you  obtain? 

Ratio,  proportion  and  variation*  Under  ’’Ratio,  pro- 
portion and  variation”,  the  National  Committee  recommends  that 
the  pupil  gain  a "clear,  working  knowledge  of  the  idea  of  ratio 
and  its  uses,  and  of  what  is  meant  by  saying  that  a variable  is 
proportional  to  another  variable”;  hence  material  of  this 
nature  is  included  under  "topics  to  be  included”;  the  meaning 
of  technical  terms,  such  as  "inversely  proportional  to"  and 
"mean  proportional “ , is  "to  be  included". 

Theorems  on  alternation,  inversion,  composition  and 
division  are  topics  "to  be  omitted”  in  accordance  with  the  sug- 
gestions of  the  National  Committee. 

Factoring.  The  Committee  recommends  that  only  these 
four  types  of  factoring  be  presented  in  algebra; 

(1^  the  monomial  type,  as  a (b4-  c)  — ab  4-  ac 

(2)  the  difference  of  two  squares,  as 

a2  - b“—  (a+-b)(a  - b) 

(3)  the  square  of  a binomial,  as 

( a t- b j 2 = a-  +-  2ab  +-  b2 

(4)  trinomials  of  the  second  degree  that  can  be 

easily  factored  by  trial,  as 

x24-2x  - 15  - (x 4-5)  (x  - 3). 

These  types  are  included  in  the  list  of  topics  "to  be 
included" . To  other  types  such  as  those  given  below  the  Com- 
mittee gives  no  consideration.  Consequently  they  are  omitted 
from  the  present  investigation. 


22 

(1)  the  common  compound  type,  as 

ax  + ay  + bx  f by  = la  + b)  (x-f  y) 

(2)  the  sum  or  difference  of  two  cubic,  as 

a?  +-  b^  — ( a t-  b)  ( a-  - ab  4-  b2 ) 

Sections  of  texts  on  "Equations  solved  by  factoring" , 
are  included  also  under  the  topic  "Factoring",  since  they  are 
given  primarily  for  practice  in  factoring. 

"Fractions"  includes  the  four  fundamental  operations 
with  fractions  and  also  that  space  given  especially  to  fractional 
equations.  The  Report  says,  "the  fundamental  operations  with 
fractions  should  be  considered  only  in  connection  with  simple 
cases."  Those  fractions  are  called  simple  which  have  whole 

numbers  or  letters  in  the  numerator  and  denominator,  as  2x“  - 2y~ ; 

5x^  5y5 

and  fractions  of  the  following  type  are  called  complex:  3ab 

x 

ba/:'b 

“x2~~ 

"Theory  of  Exponents11  is  considered  to  be  included  in 
a text  if  the  topic  of  exponents  includes  a theoretical  explana- 
tion of  negative  and  zero  exponents.  "Theory  of  quadratic  equa- 
tions" is  interpreted  to  mean  a theoretical  explanation  of  roots. 

Drill  and  Verbal  Exercises.  For  the  study  of  the  per- 
cent of  drill  and  verbal  exercises,  a criterion  was  set  for  each 
type.  The  two  types  are  referred  to  as  examples  and  problems. 

The  word  "example"  is  used  to  designate  the  drill  type  of  exercise, 
which  calls  definitely  for  certain  algebraic  operations.  The 
word  "problem"  is  used  to  designate  those  exercises  in  which  the 
pupil  is  required  to  determine  what  operations  are  to  be  performed. 


23 

This  type  requires  some  "reasoning"  on  the  part  of  the  pupil. 
Problems  are  always  stated  in  verbal  form,  while  examples  of  the 
drill  type  usually  are  not.  "Problems“  are  frequently  referred 
to  as  practical  or  verbal  exercises.  For  example,  the  following 
exercises  are  drill  exercises  or  examples: 

(1)  Add,  8ab,  -9cd,  ~6ab,  and  +-  4cd 

(2)  Solve,  x - 3 =12;  7y  = 3y  - 12 

(3^  G-et  the  temperature  readings  in  your  own  school 
district  tomorrow,  and  draw  the  graph. 

Those  following  are  problems: 

(1)  A's  age  is  three  fifths  of  B*s  age;  but  in  16 

years  A's  age  will  be  five  sevenths  of  B’s-age. 
Find  their  ages  at  present. 

(2)  The  base  of  a rectangle  is  9 feet  more  and  the 

altitude  is  8 feet  less  than  the  side  of  a square. 
The  area,  of  the  rectangle  exceeds  the  area  of  the 
square  by  15  square  feet.  Find  the  dimensions 
of  the  rectangle. 

(3)  What  number  increased  by  11  equals  19? 

With  these  criteria  each  text  was  examined,  first,  to 
discover  which  topics  and  processes  were  included  of  those  recom- 
mended by  the  Committee  "to  be  included"  and  “to  be  omitted"  and 
the  amount  of  space  given  to  them;  second,  to  discover  the  per- 
cent of  examples  and  problems;  and  third,  for  the  space  given  to 
history  and  biography.  The  examination  of  texts  included  all 
pages  except  those  given  to  the  preface,  the  index,  the  table  of 
contents,  and  pages  on  which  answers  are  printed.  Those  pages 
that  did  not  classify  under  "topics  to  be  included"  or  “topics 
to  be  omitted”  were  omitted  from  any  consideration.  A few  topics 
of  this  type  are  "Inequalities,"  “Limits,"  and  "Identities  and 
Equations  and  Condition." 


. 


- 


Chapter  III 

The  Content  of  Twelve  First  Year  Algebra  Texts 

Following  the  procedure  outlined  in  Chapter  II,  the 
content  of  the  texts  was  studied.  The  purpose  of  the  present 
chapter  is  to  present  the  material  in  each  text.  Tables  have  been 
arranged  showing  (1)  the  topics  and  space  included  which  the  Com- 
mittee recommended  "to  be  included",  (2)  topics  and  space  given 
that  are  recommended  by  the  Committee  "to  be  omitted",  (3)  the 
percent  of  drill  and  verba.1  exercises,  and  (4)  the  space  given 
to  history  and  biography. 

Topics  "to  be  included" . Table  I shows  how  many  of  the 
topics  and  processes  which  the  Committee  recommends  "to  be  in- 
cluded, “ each  text  has  included  in  its  content.  A topic  marked 
with  a plus  sign,  thus  (+) , is  included  in  the  text  named  at  the 
top  of  the  column,  while  a minus  sign,  thus  (*•),  indicates  that 
a topic  has  been  omitted  from  the  text  designated  at  the  top  of 
the  column.  We  see  from  Table  I that  the  formula,  as  a concise 
language,  was  not  included  until  1898;  the  formula,  as  an  ex- 
pression of  the  dependence  of  variables,  was  omitted  until  1908. 

No  graphing  of  any  kind  was  included  up  to  1897;  at  that  date, 
graphs  and  graphing  were  included  in  the  "appendix"  to  Wells'  text; 
in  1898  Wentworth* s text  did  not  offer  graphs,  but  after  1906  the 
topic  was  given  a place  in  the  content  of  each  text  studied. 
"Positive  and  negative  numbers"  as  expressing  magnitude  and  direc- 
tion was  given  in  Wentworth ‘ s text  published  in  1881,  but  was 


! . 


■ 


3 i~ 

Table  I 

Detailed  Table  of  Topics  Included  in  Texts 


ffent- 

Went- 

■ 

£oung 

Hawke s 

Wells 

Slaught 

Hawke s 

Dure 11 

worth 

Milne 

Wells 

worth 

Wells 

and 

Luby 

and 

Milne 

and 

Luby 

and 

Topic 

Jack- 

and 

Hart 

Lennes 

and 

Arnold 

son 

Touton 

Touton 

1881 

1392 

1897 

1898 

1906 

1908 

1910 

1912 

1915 

1915 

1917 

1920 

Formula 

4 

4 

4 

4 

a.  As  a concise  language 

b.  Shorthand  rule  of 

- 

- 

- 

4r 

4 

4 

4 

■+ 

*+ 

4 

4 

4 

+ 

computation 

4 

- 

4- 

4- 

4 - 

4 

c . As  a general  solution 

d.  As  expression  of  the 

■r 

• 

4- 

4- 

4- 

4 

4 

4 

4 

4 

-4” 

dependence  of 

4 

variables 

- 

- 

- 

- 

- 

+ 

— 

+* 

4 

4 

Graph 

a.  As  a method  of  repre- 

4 

+ 

4 

4 

senting  facts 

- 

- 

■h 

- 

b.  As  a method  of  repre- 

4 

presenting  dependence 

- 

- 

4 

- 

4- 

4- 

*+- 

4 

4 

4 

4 

c . As  a method  of  solv- 

ing  problems 

- 

4- 

"V" 

4 

4 

4 

Positive  and  negative  numbers 

a.  Expressing  magnitude 

4- 

4 

4 

4 

4 

and  direction 

-V 

- 

4- 

4- 

4 

b.  Graphic  representation 

- 

- 

- 

- 

4 

4 

4 

4 

4 

4 

4 

c.  Fundamental  operations 

4 

-4- 

-+- 

't 

i~ 

4 

“4* 

4 

4 

4 

4 

The  Equation 

a.  Linear  quadratic  in 

4- 

-4 

4 

one  unknown 

-t 

+ 

4 

4 

4 

4 

4 

4 

b.  Equations  in  two 

4- 

variables : 

4 

4- 

4 

4 

4 

4 

4 

4 

-4 

1 As  expressing  a func 

tional  relation 
2 Determination  of  un 
knowns 

4 

4- 

4- 

"4 

4 

4 

4 

4 

4 

4 

4 

c.  Ratio,  proportion  and 

-4- 

4- 

4 

4- 

4 

variation 

4 

4 

4 

4 

4 

4- 

4 

Algebraic  technique 

. 4- 

4- 

a.  Fundamental  operations 

b.  Factoring 

4 

4- 

4 

4 

4 

4 

4 

4 

-f 

1 Monomial 

2 Difference  of  two 

4 

4- 

-f 

4" 

4- 

4“ 

i 

4 

4 

4 

4 

4 

4 

4- 

squares 

\ 

4r 

4 

4 

4* 

r 

4 

3 Square  of  binomial 

T 

4 

f 

*T" 

4 

4- 

by  trial 

4- 

4 

4 

4- 

4 

4 

■f 

-f- 

~ir 

4 

— f' 

c.  Fractions 

d.  Exponents  and 

4: 

4- 

4 

4 

4 

4 

4 

4 

4 

4 

4 

-v 

"4~ 

4 

4 

radicals 

-t- 

4- 

r 

4 

4 

4 

4 

Number  of  topics  included 

15 

12 

18 

16 

20 

20 

20 

2 1 

20 

2 1 

21 

21 

Percent  of  topics  included 

71 

57 

86 

76 

95 

95 

95 

10U 

95 

100 

100 

100 

26 

omitted  from  Milne's  text  in  1892;  in  1897  the  topic  was  again 
included  and  is  given  in  all  texts  published  later  than  that  date. 
Graphic  representation  of  positive  and  negative  numbers  was 
omitted  until  1906  and  since  then  it  has  been  offered.  The  re- 
maining topics,  as  the  checks  in  Table  I indicate,  are  all  in- 
cluded in  all  t?/elve  texts  that  were  examined. 

At  the  bottom  of  each  column  a summary  has  been  made  of 
the  number  of  topics  each  text  contains,  which,  in  accordance  with 
the  Committee's  recommendations,  should  be  given.  Reading  the 
column  headed  "Wentworth  1381“  as  an  example,  we  see  that  in 
Wentworth*  s text  published  in  1881  fifteen  topics  of  the  possible 
twenty-one  are  included,  and  in  the  next  column  twelve  of  the 
twenty-one  are  given.  The  lowest  line,  "Percent  of  topics  in- 
cluded," indicates  that  seventy-one  percent  of  these  topics  were 
included  in  1381,  fifty-seven  percent  in  1892,  eighty-six  percent 
in  1397,  and  so  on. 

This  table  (Table  I).  does  not  mean  that  only  those 
topics  and  subtopics  that  are  approved  by  the  Committee  are  in- 
cluded; it  simply  indicates  that  at  least  these  topics  are  given. 
For  example,  under  "Exponents"  and  "Radicals”  the  Committee  re- 
commends that  "the  laws  for  positive  integral  exponents  should  be 
included,"  and  that  "the  consideration  of  radicals  should  be  con- 
fined to  the  simplification  of  expressions  involving  the  radical 
sign."  In  almost  every  case,  topics  more  advanced  and  more  com- 
plex than  these  are  included,  but  the  check  in  the  table  remains 
the  same  as  if  only  the  required  topics  and  processes  were  given 
in  the  text.  In  Table  III  and  Table  (a  table  of  conclusions) 


27 


attention  is  given  to  the  amount  of  space  given  to  topics  and  pro- 
cesses that  are  unnecessary. 

Table  II  showing  the  space  given  to  topics  which  the 
Committee  recommends  u to  be  included"  is  more  explicit.  This 
table  shows  In  terms  of  the  total  space  in  the  text  the  percent 
of  space  that  each  text,  indicated  at  the  top  of  the  column,  has 
given  to  the  topic  indicated  in  each  line.  Thus,  Wentworth  (pub- 
lished in  1881)  gives  .07  percent  of  the  total  space  in  the  text 
to  the  formula.  This  text  gives  no  material  on  the  graph,  .9  per- 
cent to  positive  and  negative  numbers,  2.8  percent  to  linear  equa- 
tions in  one  unknown,  and  so  on  down  the  column.  Reading  the 
lines  from  left  to  right,  we  find  .07  percent  of  the  space  in  the 
text  was  given  to  the  formula  in  1881,  none  in  1892,  and  .2  per- 
cent in  1897;  thereafter  the  space  given  to  the  formula  varies 
irregularly  between  1 percent  and  4.9  percent  of  the  total  space 
in  the  text.  The  "graph",  which  was  not  included  until  1897,  was 
given  2.3  percent  of  space  at  that  date,  but  was  omitted  entirely 
in  1898.  After  1906  the  space  given  to  this  topic  varied  irregular- 
ly from  2 percent  in  1906  to  10  percent  in  1920.  Pointing  out 
the  changes  in  texts  where  revised  editions  have  been  published, 
we  can  make  no  general  conclusion  with  respect  to  "graphs" : 
neither  edition  of  Wentworth  contains  any  material  on  the  graph; 
the  first  edition  of  Milne  ( 1892)  gives  none  of  its  space  to  graphs 
and  the  later  edition  (1915)  gives  7.8  percent;  Wells  in  1897 
gives  2.3  percent  space  to  the  graph,  in  1906,  2 percent,  and  in 
1912  in  the  text  revised  by  Wells  and  Hart  5.9  percent  of  the 
space  was  devoted  to  graph. 


t 


, 


. 


Tab] 

Detailed  Table  Show] 
Unnecess 

1 

.e  III 

.ng  Sp 
jary  T 

ace  Given  to 
opics 

i .. 

Went- 

Went- 

Young 

Hawke s 

Wells 

SI aught 

Hawke s 

Durell 

worth 

Milne 

Wells 

worth 

Wells 

and 

Luby 

and 

and 

Luby 

and 

Topics 

Jack- 

and 

Hart 

Milne 

Lennes 

and 

Arnold 

son 

Touton 

Touton 

1881 

1892 

18*7 

1898 

1908 

1908 

1910 

1912 

191o 

.1915 

1917 

1920 

Highest  common  factor  and  lowest 

* 

' 

- 

* • 

common  multiple 

4.3 

2.y 

4.0 

3.o 

1.2 

2.0 

1.0 

1.9 

1.6 

1 .4 

1.4 

.4 

Proportion 

. o 

.9 

.4 

.4 

' .3 

0 

.9 

.7 

.4 

.3 

1 .2 

0 

Literal  equations 

3.  i 

.7 

1.8 

1.8 

1.9 

.2 

1.6 

1 .5 

1 .4 

.7 

2.9 

1.5 

Radicals 

3.3 

5.4 

2.3 

4.0 

4.5 

1 . 6 

4.y 

0 

2.4 

2.4 

5.  1 

4.3 

Square  root  of  polynomials 

• 6 

.8 

.8 

1 .0 

1 .0 

.8 

1.0 

.7 

.8 

1 . 1 

.9 

.6 

Cube  root 

1.5 

.8 

2.0 

1.1 

1.8 

1 o 

.2 

.4 

.3 

0 

.5 

.07 

Theory  of  exponents 

1.2 

2.  1 

1.9 

.8 

2.2 

.3 

.3 

.2 

.2 

0 

0 

0 

Simultaneous  equations  in  more 

than  two  unknowns 

.8 

1.9 

1 .8 

1. 1 

2.9 

2 . 0 

1 .0 

1.0 

.9 

1 .4 

1.3 

.6 

Pairs  of  simultaneous  quadratic 

equations 

1.5 

1.7 

2.3 

1.7 

1.7 

0 

.2 

0 

1 .2 

1 .2 

.8 

0 

Theory  of  quadratic  equations 

2.2 

u 

1.4 

2.8 

0 

0 

0 

0 

0 

.08 

0 

Binomial  theorem 

2.8 

5.4 

1.8 

4.9 

2.6 

.9 

0 

0 

.9 

1.7 

0 

0 

Progressions 

4.0 

4.8 

5.  1 

3.9 

4.4 

U 

0 

0 

0 

0 

0 

0 

Theory  of  imaginary  and  complex 

numbers 

.3 

0 

0 

1 .4 

0 

0 

0 

0 

0 

0 

0 

0 

Radical  equations 

1 .6 

1.0 

.5 

.9 

.6 

' *07 

1.7 

0 

2.4 

.5 

.5 

0 

Equations  of  degree  higher  than 

second 

.4 

.7 

.0 

1 . 1 

.7 

0 

.07 

0 

.7 

.5 

.08 

.3 

Total  pages 

325 

367 

357 

407 

458 

323 

329 

321 

334 

352 

325 

335 

Total  percent 

28.6 

,28.7 

25. 1 

29.  1 

28.6 

10.  17 

13.7 

6.4 

13.2 

1 1 .2 

14.7 

7.8 

29 


The  space  given  to  "positive  and  negative  numbers"  in- 
creases from  .9  percent  in  1 38 1 to  2.4  percent  in  1920,  with  two 
higher  percents  of  space  given  to  it  in  1915s  9.6  percent  in  one 
text  and  6.3  percent  in  another. 

Reading  the  two  extremes  in  dates  we  see  that  the  space 
given  to  the  "linear  equation  in  one  unknown"  increases  from  2.8 
percent  in  1381  to  11  percent  in  1920.  This  increase  is  a gradual 
one  showing  at  only  one  place  a percent  of  space  lower  than  in 
1881.  This  exception  is  in  1906  when  2.6  percent  were  given. 
"Quadratic  equations  in  one  unknown"  is  included  in  every 
text,  and  is  given  an  irregular  increase  in  space.  With  the  ex- 
ception of  the  text  published  in  1381  "linear  equations  in  two 
variables"  is  given  more  space  in  the  six  later  published  texts 
than  in  the  older  ones.  One  text  ( 1908)  fails  to  include  "quad- 
ratic equations  in  two  unknowns."  Noting  the  two  extremes  in 
dates,  we  observe  that  the  oldest  text  (1381)  gives  just  as  much 
space  to  this  topic  as  the  newest  one;  the  greatest  percent  of 
space  is  2,3  in  1915*  Treatment  of  the  topic  "exponents"  is  ex- 
cluded from  one  text  (1892).  The  line  marked  "total  pages"  is 
read  in  the  following  manner:  Wentworth  ( 1 83 1 ) consists  of  325 

pages  of  mathematical  material  exclusive  of  pages  devoted  to  a 
preface,  table  of  contents,  index,  and  answers.  The  line  begin- 
ing  "total  percent"  points  out  that  47  percent  of  the  total  space 
in  the  text  has  been  given  to  topics  and  processes  that  ought  to 
be  included. 

Topics  "to  be  omitted."  For  the  purpose  of  showing 


the  content  of  each  text  that  is  "to  be  omitted,"  as  the  National 


1 


- 


. ■ 

. • ■.  :..i  ■ • ii. 


t 


30 


Committee  recommends,  Table  III  has  been  arranged.  Using  the  first 
column  as  an  example,  information  presented  there  is  read  in  this 
manner:  Wentworth's  text  published  in  1881  gives  4.3  percent  of 

its  total  space  to  "highest  common  factor"  and  "lowest  common 
multiple";  .6  percent  of  the  space  is  given  to  proportion,  3.1 
percent  to  literal  equations  and  so  on  down  the  column  to  the  last 
item  indicating  that  .4  percent  of  the  texts  space  is  given  over 
to  equations  of  degree  higher  than  second.  In  the  following  line, 
we  have  given  the  total  number  of  pages  in  the  text,  325  in  this 
case;  here,  as  in  Table  II  "total  number  of  pages"  excludes  the 
preface,  table  of  contents,  index  and  answers.  The  line  beginning 
"total  percent  of  space  that  should  be  omitted"  indicates  that 
28.6  percent  of  the  space  is  devoted  to  topics  that  are  "to  be 
omitted"  in  accordance  with  the  suggestions  recommended  in  the 
Report . 

Observing  the  first  line  in  Table  III  a gradual  decrease 
is  noted  in  the  space  given  to  "highest  common  factor  and  lowest 
common  multiple"  from  4.3  percent  in  1881  to  .4  percent  in  1920. 
The  next  two  topics,  "proportion"  and  "literal  equations"  also 
show  a decrease.  The  space  given  to  "radicals"  increases  from 
3.5  percent  in  1881  to  4.3  percent  in  1920.  However,  lower  per- 
cents are  shown  at  dates  previous  to  1920,  for  example,  Wells  and 
Hart's  text  (1912)  contains  no  "radicals"  to  be  omitted,  Young 
and  Jackson  in  1908  gives  1.6  percent,  and  the  two  texts  in  1915 
each  give  2.4  percent  of  their  space  to  "radicals."  "Theory  of 
exponents"  loses  space  gradually  and  after  1915  we  find  the  topic 
omitted  entirely.  Three  texts,  those  published  in  1908,  1912,  and 


: 


* 


« • • 


, 


. 


; 


. 


High* 

Propc 

Lite} 

Radic 

Squa: 

Cube 

Theo} 

SimiL 

Pairi 

Theo: 

Binoi 

Prog: 

Theo; 

Radi' 

Equa 

Tota 
Tot  a 


1 


82 

192G  omit  "pairs  of  simultaneous  equations."  Four  of  the  twelve 
texts  examined  included  "theory  of  quadratic  equations,"  one  of 
these  texts  was  published  as  recently  as  1917*  The  two  texts 
published  in  1915  give  .9  percent  and  1.7  percent  of  their  space 
to  the  "binomial  theorem",  while  four  texts,  published  in  1910, 

1912,  1917  and  1920  omit  this  topic.  "Progressions"  is  excluded 
after  1996.  Wentworth's  two  texts  are  the  only  texts  including 
the  "theory  of  imaginary  and  complex  numbers",  making  1898  the 
latest  date  when  this  topic  was  included.  Only  two  texts  omit 
"radical  equations"  as  the  National  Committee  recommends?  these 
two  were  published  in  1912  and  I92O.  "Equations  of  degree  higher 
than  second"  is  included  in  every  text  but  two  ( 1908  and  1912;. 

The  figures  in  this  line  show  no  general  decrease  in  the  space 
given  to  this  topic . 

Drill  and  Verbal  Exercises.  From  the  summary  of  the 
Report  of  the  National  Committee  (Chapter  1),  it  has  been  ob- 
served that  algebra  should  include  some  exercises  of  each  of  two 
types  (1)  drill  and  (2)  verbal  exercises.  Table  IV  shows  the  re- 
sults of  such  a study  of  the  numbers  of  drill  and  verbal  problems 
in  each  text.  Using  the  line  beginning  Wentworth  (1881)  as  il- 
lustrative, the  number  of  drill  and  verbal  exercises  is  pointed 
out  as  follows:  Wentworth's  text  published,  in  1831  contains 

2158  examples,  and  342  problems,  or  verbal  exercises;  this  placed 
in  percents  reads  86  percent  drill  and  14  percent  verbal  exercises 
are  given  in  this  text.  The  next  line  shows  4038  drill  and 
549  verbal  exercises  in  Milne's  text  published  in  1882;  or  changed 
to  percents,  this  is  88  percent  drill  and  12  percent  verbal  exercise 


33 


Table  IV 

Drill  and  Verbal  Exercises  in  Texts 


1 

f=  — “ 

Text 

Date 

Number  of  exercises 

Percent 

of  exercises 

Drill 

Practical 

Drill 

Practical 

Wentworth 

1881 

2158 

342 

86 

14 

Milne 

1892 

4058 

549 

88 

12 

Wells 

1897 

3027 

328 

90 

10 

Wentworth 

1898 

3015 

475 

86 

14 

Wells 

1906 

3296 

335 

69 

1 1 

Young  and  Jackson 

1908 

4228 

464 

90 

10 

Hawke s,  Luby  and 
Touton 

1910 

3077 

61  1 

64 

16 

Wells  and  Hart 

1912 

2932 

537 

85 

15 

Milne 

1915 

3815 

661 

85 

15 

SI aught  and 
Lennes 

1915 

4118 

662 

86 

14 

Hawke s,  Luby  and 
Touton 

1917 

3276 

661 

83 

17 

Durell  and  Arnold 

1920 

3333 

t- 

00 

c\ 

80 

20 

, 


■ 


Pages  of  hi 
bio^ 

Page  pic  tui 

Total  pages 
and  bi< 

Percent  of 


* Portrait 
6 3/4  pages 


35 


The  fa,ct  that  no  history  or  "biography  was  given  in  any 
of  the  texts  studied  up  to  1910  is  shown  in  Table  V.  Using  the 
column  headed  Hawkes,  Luby,  and  Touton  (1910)  as  an  illustration, 
we  read  the  table  thus:  The  edition  of  Hawkes,  Luby  and  Touton, 

published  in  1910,  contains  twelve  and  one  half  pages  of  history 
and  biography,  and  six  pictures  of  mathematicians,  making 
eighteen  and  one  half  pages  in  all  of  historical  material.  This 
material  occupies  5.6  percent  of  the  total  space  of  the  text. 

In  this  tabulation  a page  picture  was  considered  first  as  a topic 
in  itself  and  later  combined  in  the  total  number  of  pages  and  the 
percent  of  space  given  to  history  and  biography. 

These  five  tables  present  in  detail  the  content  of  the 
twelve  texts  studied.  The  next  step  in  this  investigation  is  to 
sum  up  the  tables  for  drawing  conclusions  as  to  the  trend  of  the 
content  of  algebra,  texts. 


Chapter  XV 

Summary:  The  Trend  of  the  Content 

The  purpose  of  Chapter  IV  is  to  sum  up  the  information 
tabulated  in  Chapter  III  into  a form  such  that  the  trend  of  the 
content  of  the  twelve  texts  may  he  observed.  Table  VI  is  a sum- 
mary of  the  five  preceding  detailed  tables;  each  column  gives 
the  information  secured  concerning  the  total  content  of  the  text 
named  at  the  top  of  the  column;  for  example,  the  first  column 
points  out  that  Wentworth ( 1881 ) contains  (1;  71  percent  of  the 
21  topics  recommended  bjr  the  Committee  ‘‘to  be  included”,  and  (2) 
all  of  the  15  topics  that  ought  "to  be  omitted";  (3J  46.9  per- 
cent of  the  total  space  in  the  text  is  given  to  topics  and  pro- 
cesses that  should  "be  included",  (4;  28.6  percent  of  the  total 
space  is  given  to  topics  and  processes  that  ought  to  be  omitted, 
according  to  the  National  Committee,  (5)  86  percent  of  all  the 
exercises  in  this  text  fall  under  the  drill  type  of  exercise, 
and  the  other  14  percent  are  exercises  of  the  verbal  type;  (6)  no 
history  or  biography  is  given  in  this  text.  Other  columns  are 
read  in  a similar  manner. 

These  same  results  are  presented  also  in  graphical 
form.  Figure  1 shows  the  percent  of  topics  and  the  percent  of 
space  given  to  topics  that  are  "to  be  included"  from  1881  to 
1920;  Figure  2 shows  the  percent  of  topics  and  processes  that  are 
"to  be  omitted"  and  the  percent  of  space  given  to  these  topics 
during  the  same  years;  and  Figure  3 indicates  graphically  the 


. 


. 


>• 


■j 


. 

• 

■ 

( 1 ) Perce 
thal 

(2)  Perce 

thal 

(3)  Perce 

thal 

(4)  Perce 

thai 

(5;  Perce 

(6)  Perce 

( 7 ) Perce 

torj 


3? 

Table  VI 

Summary  of  Detailed  Tables 


Topics  of  Tables 

Went- 

worth 

1881 

Milne 

1892 

Wells 

1897 

Went- 

worth 

1898 

Wells 

1906 

foung 

and 

Jack- 

son 

1908 

Hawke s 
Luby 
and 

Touton 

1910 

Wells 

and 

Hart 

1912 

Milne 

1915 

SI aught 
and 

Lennes 

1915 

Hawke s 
Luby 
and 

Touton 

1917 

Durell 

and 

\rnold 

1920 

(U 

Percent  of  topics  included 
that  should  "be  included" 

-71 

57 

86 

76 

95 

95 

95 

100 

95 

100 

100 

100 

( 2 ) 

Percent  of  topics  included 
that  should  "be  omitted" 

100 

b7 

87 

100 

93 

60 

73 

47 

73 

67 

73 

47 

(3) 

Percent  of  space  to  topics 
that  are  "to  be  included" 

46.9 

47.3 

41. 1 

50.2 

32.0 

52.8 

62.5 

68.4 

60.4 

50.2 

64.5 

68.7 

(4) 

Percent  of  space  to  topics 
that  are  "to  be  omitted" 

28.6 

28.7 

25.1, 

29.2 

28.6 

10.2 

14.4 

6.4 

10.3 

10.1 

15.8 

7.7 

(5) 

Percent  of  drill  exercises 

06 

88 

90 

86 

89 

90 

34 

85 

85 

86 

83 

80 

(6) 

Percent  of  verbal  exercises 

14 

12 

10 

14 

1 1 

10 

16 

15 

15 

14 

17 

20 

(7) 

Percent  of  space  to  his- 
tory and  biography 

0 

0 

0 

0 

0 

0 

5.6 

1 .2 

.2 

4.4 

4.4 

0 

percent  of  drill  and  verbal  exercises  included  in  the  texts 
examined.  From  a study  of  Table  VI  and  these  graphs  we  may  as- 
sume conclusions  concerning  the  trend  of  the  content  of  algebra 
texts . 

Topics  and  processes  "to  be  included."  Table  VI 
shows  the  range  in  percents  of  space  given  to  topics  and  pro- 
cesses "to  be  included"  from  .369  to  .750.  The  lowest  percent 
in  the  six  latest  published  texts  is  higher  than  the  highest 
percent  found  in  the  six  older  texts.  From  1681  to  1920  we  find 
a change  in  the  space  given  to  topics  "to  be  included"  from 
46.9  percent  to  72.1  percent  with  the  two  extremes,  36.9  percent 
and  75.0  percent  falling  at  1905  and  1917  respectively.  The 
upper  portion  in  Figure  1 represents  the  percent  of  topics  in- 
cluded of  the  twenty-one  topics  which  the  National  Committee 
recommends  "to  be  included."  This  line  also  shows  a rise  from 
1881  to  1920  almost  parallel  to  the  rise  in  the  percent  of  space 
that  should  be  included;  in  1861,  71  percent  of  these  topics 
and  processes  were  included,  while  in  1920,  100  percent  were 
given.  Summing  up  the  facts  concerning  the  topics  and  processes 
that  are  "to  be  included",  a decided  increase  is  noted  in  (1) 
the  number  of  topics  and  processes  included,  and  in  (2)  space 
given  to  these  topics  and  processes. 

Topics  and  processes  "to  be  omitted."  Table  VI  in- 
dicates that  28,6  percent  of  the  total  space  in  the  text  pub- 
lished  in  1881  was  given  to  material  which  the  Committee  con- 
siders unnecessary,  in  1698  we  find  29.2  percent,  and  in  1908, 

10  percent  of  the  page  were  used  for  non-essential  matter;  in 


q+b 

*nj  r 

-ff 

+ 

42 


1912,  6.4  percent,  and  in  1920,  7.7  percent.  Thus  we  see  the 
two  extremes,  represented  in  189b  and  1912,  are  29*2  percent  and 
6.4  percent.  The  lower  line  in  Figure  2 represents  these  results: 
the  greatest  percent  of  unnecessary  material  in  1898,  and  the 
least  in  1912.  Even  though  these  variations  cause  the  graph  to 
he  irregular  from  1881  to  1920  there  is  a decrease  of  20.9  per- 
cent indicated  in  the  amount  of  material  presented  that  ought 
"to  be  omitted."  A comparison  of  the  two  graphs  represented  in 
Figure  2 shows  that  the  percent  of  topics  "to  be  omitted"  has 
decreased  along  with  the  decrease  in  the  percent  of  space  given 
to  topics  that  should  be  omitted.  The  upper  line  in  Figure  2 
shows  a decrease  of  53  percent  in  the  number  of  topics  included; 
in  1881,  100  percent  of  the  topics  were  included  that  the  National 
Committee  recommends  to  be  omittedJ  while  in  1920  only  47  percent 
were  given  in  the  text.  In  conclusion,  it  is  noted  that  the 
number  of  topics  "to  be  omitted"  has  decreased  53  percent  and 
that  the  space  given  to  these  topics  has  decreased  20.9  percent. 

Drill  and  Verbal  Exercises.  The  lines  in  Table  VI, 
indicating  the  percent  of  drill  and  verbal  exercises  in  each 
text,  show  a rather  marked  agreement  among  authors  as  to  the 
ratio  of  the  two  types  of  exercises.  The  two  extremes  are  in- 
dicated in  the  texts  published  in  1397  and  1908  (both  giving  the 
same  percent  of  drill  and  verbal  exercises)  and  in  the  text 
published  in  1920.  In  1897  and  1908  we  find  9p  percent  of  all 
the  exercises  are  examples  and  10  percent  are  problems.  In  1920, 
80  percent  of  the  total  number  of  exercises  are  examples  and 


. 

« 


43 


20  percent  are  problems.  Taking  the  two  extremes  in  dates,  we 
find  that  in  1881  86  percent  are  drill  examples,  and  14  percent 
are  problems;  and  in  1920,  as  quoted  above,  we  have  80  percent 
drill  and  20  percent  verbal  exercises.  Reading  only  these  two 
results,  we  might  say  that  there  is  a little  tendency  to  have  our 
texts  contain  less  of  drill  and  mor*e  of  application  material;  it 
might  be  said  that  our  texts  are  becoming  more  “wordy" , as  the 
National  Committee  recommends.  Dividing  the  texts  examined  into 
two  classes,  the  older  six  and  the  newer  six,  as  before,  we 
might  also  say  that  there  is  a slight  tendency  for  the  newer 
texts  to  give  fewer  drill  examples  and  more  verbal  problems  than 
the  older  texts.  Figure  3 indicates  the  percent  of  drill  and 
practical  exercises  in  the  year  indicated  on  the  lower  line. 

This  graph  points  out  that  the  number  of  drill  exercises  is 
always  at  least  60  percent  greater  than  the  number  of  verbal 
exercises.  The  upper  line  shows  a decrease  of  6 percent  in  the 
number  of  drill  exercises  included  in  the  texts;  the  lower  line 
indicates  the  percent  of  practical  exercises,  and  shows  an  in- 
crease of  6 percent  in  the  number  of  practical  exercises  given. 
Summarizing  the  content  of  the  twelve  texts  we  would  say  that 
the  trend  of  algebra  texts  is  slightly  toward  the  suggestion  me.de 
by  the  Committee:  "Continued  emphasis  throughout  the  course 

must  be  placed  on  the  development  of  power  in  applying  ideas, 
processes,  and  principles  to  concrete  problems,  rather  than  the 
acquisition  of  new  facility  or  skill  in  manipulation." 

History  and  biography.  These  topics  have  been  in- 
cluded in  texts  only  recently*  it  is  observed  in  Table  VI  that 


44 


up  to  1910  no  history  or  biography  was  given  in  any  text.  The 
text  published  in  19 10,  however,  gave  5.6  percent  of  its  space 
to  these  topics;  in  I9I2,  1,2  percent  was  given  to  history  and 
biography;  in  I9I5  one  text  shows  .2  percent  and  another  4,4 
percent;  and  in  1920  no  history  or  biography  was  given.  Dis- 
tinguishing between  the  six  older  and  the  six  newer  texts,  we 
may  say  that  the  newer  texts  tend  to  include  more  history  and 
biography  than  the  older  ones.  However,  there  is  no  gradual  in- 
crease evident  in  the  amount  of  space  given  to  these  topics. 

Concluding  statement.  Regarding  the  preliminary  report 
of  the  National  Committee  on  Mathematical  Requirements  as  repre- 
sentative of  the  material  that  should  be  taught  in  the  secondary 
schools  of  today,  and  basing  our  conclusions  on  this  examination 
of  twelve  first-year  algebra  texts,  the  following  statements  are 
made  concerning  the  trend  of  the  content  of  first-year  algebra 
texts  since  1381  : There  is  a tendency  to?/ard  omitting  and  giving 

less  space  to  topics  and  processes  which  the  National  Committee 
recommends  "to  be  omitted."  There  is  a tendency  toward  including 
and  giving  more  space  to  those  topics  and  processes  which  the 
National  Committee  recommends  be  included.  The  percent  of  drill 
exercises  tends  to  decrease  slightly,  and  the  percent  of  verbal 
exercises  increases  slightly  indicating  a relative  increase  in 
wordy  exercises  as  the  Committee  recommends.  The  tendency  to 
include  history  and  biography  is  slight,  but  gives  evidence  of 
some  thinking  in  historical  terms. 


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